Sistemas y Señales Biomédicos

Ingeniería Biomédica

Ph.D. Pablo Eduardo Caicedo Rodríguez

2025-07-23

Sistemas y Señales Biomedicos - SYSB

Digital Filter – Introduction

  • It is a mathematical algorithm or system that processes digital signals.
  • They enhance, suppress, or modify specific frequency components.
  • These filters are essential for removing noise, extracting relevant information, and improving signal quality.

Digital Filter – Introduction

Importante

The digital filter separates the noise and the information of a discrete signal.

Digital Filter – Introduction

Digital Filter – Introduction

Suppose a discrete time system \[ y[n] = \sum_{k=1}^{K} a_k y[n - k] + \sum_{m=0}^{M} b_m x[n - m]\]

  • K y M are the order of the filter.

  • We must know the initial condition.

Examples of digital filters

Gain

\[y[n] = G x[n]\]

Delay of \(n_0\) samples

\[y[n] = x[n - n_0]\]

Two points moving average

\[y[n] = \frac{1}{2} (x[n] + x[n - 1])\]

Euler approximation of the derivative

\[y[n] = \frac{x[n] - x[n - 1]}{T_s}\]

Averaging over N consecutive epochs of duration L

\[y[n] = \frac{1}{N} \sum_{k=0}^{N-1} x[n - kL]\]

Trapezoidal integration formula

\[y[n] = y[n - 1] + \frac{T_s}{2} (x[n] + x[n - 1])\]

Digital “leaky integrator” (First-order lowpass filter)

\[y[n] = a y[n - 1] + x[n], \quad 0 < a < 1\]

Digital resonator (Second-order system)

\[y[n] = a_1 y[n - 1] + a_2 y[n - 2] + b x[n], \quad a_1^2 + 4a_2 < 0\]

The impulse response

  • The impulse response, denoted as \(ℎ[n]\), is the output of a digital filter when the input is a unit impulse function \(\delta[n]\)
  • The impulse response fully describes the system. Given \(h[n]\), we can determine the output for any input using convolution.
  • Different types of filters (low-pass, high-pass, band-pass, etc.) have characteristic impulse responses.

Conditions

For a system’s response to be fully described by its impulse response, the system must satisfy the following key conditions.

Linearity

If the system responds to \(x_1[n]\) with \(y_1[n]\) and to \(x_2[n]\) with \(y_2[n]\), then:

\[y[n] = y_1[n] + y_2[n]\]

Homogeneity

If the input is scaled by a constant \(c\), the output is also scaled:

\[\text{If } x[n] \rightarrow y[n], \text{ then } cx[n] \rightarrow cy[n]\]

Time Invariance

A system must be time-invariant, meaning a time shift in the input causes the same shift in the output:

\[\text{If } x[n] \rightarrow y[n], \text{ then } x[n - n_0] \rightarrow y[n - n_0]\]

Causality

A causal system is one where the output at time \(n\) depends only on present and past inputs:

\[h[n] = 0 \quad \forall n < 0\]

Stability

If the impulse response does not satisfy this condition, the system may produce unbounded outputs.

\[\sum_{n=-\infty}^{\infty} |h[n]| < \infty\]

Convolution Representation

If all condition met then \[y[n] = x[n] * h[n] = \sum_{m=-\infty}^{\infty} x[m] h[n - m]\]

Convolution